Characteristic Subgroup

Definition

If $G$ is a group, then $H$ is a characteristic subgroup of $G$ (written as $H{⊲}_{\mathrm{char}}G$) if every automorphism of $G$ maps $H$ to itself.

Proposition

If $K{⊲}_{\mathrm{char}}H$ and $H⊲G$, then $K⊲G$.

quotient-transitive

Suppose $H⩽K⩽G$ are subgroups such that $H$ is a characteristic subgroup of $G$, and $K/H$ is a characteristic subgroup of $G/H$. Then, $K$ is a characteristic subgroup of $G$.

transitive

Suppose $H⩽K⩽G$ are groups such that $H$ is a characteristic subgroup of $K$, and $K$ is a characteristic subgroup of $G$. Then, $H$ is a characteristic subgroup of $G$.

centralizer-closed

Suppose $G$ is a group and $H$ is a characteristic subgroup of $G$. Then, the centralizer $C_G(H)$ of $H$ in $G$ is also a characteristic subgroup of $G$.

commutator-closed

Members of the derived series of a group are characteristic subgroups.

Characteristic Simple Group

怎么感觉之前那个证明 Minimal Normal Subgroup and Maximal Normal Subgroup 被绕过去了，好歹毒？

这一堆之前写过了，在这里 Characteristically Simple, 合并一下。